Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Figa-Talamanca-Herz algebras

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:329

Article

Runde, Volker

关键词: amenability;    bounded approximate identity;    coset ring;    Figa-Talamanca-Herz algebra;    locally compact group;    smooth banach space;    ultrapower;    uniform convexity;    uniformly bounded representation;   

DOI  :  10.1016/j.jmaa.2006.07.007

来源: Elsevier

【 摘 要 】

Let G be a locally compact group, and let R(G) denote the ring of subsets of G generated by the left cosets of open subsets of G. The Cohen-Host idempotent theorem asserts that a set lies in R(G) if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We apply our Cohen-Host type theorems to the study of the Figa-Talamanca-Herz algebras A(p) (G) with p E (1, infinity). For arbitrary G, we characterize those closed ideals of A(p) (G) that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those G such that A(p)(G) is 1-amenable for some-and, equivalently, for all-P E (1, infinity): these are precisely the abelian groups. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

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